Thanks everyone for your input. So here's where we are now...

Before leaving work yesterday, I used some plastic ties to secure the bottom, front corners of the baskets to the seat stays and used my bungee cords to keep the load at the front of the baskets. Big improvement. The handlebars didn't feel like they were trying to jump out of my hands the whole time.

As I left work, there was a pretty stiff breeze coming from the north. I was going south, so no problem. Heck, my average speed was two mph faster than usual! I thought, "I'll be home in no time!" About seven miles into the ride, I was coming off a red light when pedaling the bike became extremely difficult. Once across the intersection, I started looking at my rack to see if a strap or something had fallen out and gotten into the wheel. Nothing. I pedaled slowly around the parking lot, trying to find the culprit. When I tried to pedal quickly, I got more resistance than when I pedaled slowly. My RD was clicking and clacking as the chain didn't want to stay on its gear. Any ideas out there?

Yep, the freewheel side of the rear axle had slipped in the dropout. The reason? The basket mount. Notice in the image, how it sits between the dropout and the bolt.

(It's highlighted in yellow and I had to use a bolt that is narrower than a standard axle bolt)


I got out my wrench, put the axle back in place and tightened it down. I didn't make it a quarter mile before it started slipping again, so I stopped again. This time I flipped the bike over and used a wrench on each side of the wheel. As I'm standing there, hunched over my inverted bike, it started to rain. Not a spring shower, mind you, but torrents of water are pouring down on me. I finished up and righted the bike. Before I even got my feet clipped in, the axle slipped again.

I dejectedly walked my bike over to the mail center of a nearby apartment complex in search of a pay phone. There was none, but a kind woman who was checking her mail let me call my wife on her cell phone. As I was standing there in my helmet by my bike, the apartment's residents were getting home from work and coming up in to check their mail. It's not an exaggeration that 8 out of 10 people who came in made a comment equivalent or at least similar to: "You're not going to let a little rain stop you, are you?"

I got home and began fashioning supplemental supports for the baskets. Several hours of trying different materials, brought me to a torn quad chair that I saved in case I could use any of its parts
A little drilling along some Dremmel tool cutting and I had new supports for my baskets.

Time for installation. I got the baskets all lined up and went to put on the axle bolts. I couldn't get them to tighten to save my life. This is when I realized that the threads on the axle were stripped. I cannibalize the rear wheel of an old junker bike to replace the axle and began trying to figure out how I could mount the baskets without attaching them to the axle. A whole lot of head scratching and thumb twiddling and I went with rubberized cable clamps.
I put the clamps around the base of the seat stays and attached the supports. That's when I looked at my watch and realized it was 4:00am. I get up at 5:30am to get ready for work. Needless to say, I overslept and had to drive to the office. My wife did ask me, before I left if I got the bike fixed. I told her that since I dozed off a time or two while working on it, I should probably give everything a once-over before riding it to work on Wednesday.

So I don't know if my supports worked or not. I'll give it a thorough test ride when I get home tonight.

Thanks again to everyone for the tips and suggestions.